LMFDB - Embedded newform 2100.2.k.c.1849.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,2,Mod(1849,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,-4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.2.k.c.1849.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +4.00000i q^{13} -2.00000i q^{17} -2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} +2.00000i q^{33} -10.0000i q^{37} +4.00000 q^{39} -10.0000 q^{41} +12.0000i q^{43} +8.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +2.00000i q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +12.0000i q^{67} +4.00000 q^{69} -10.0000 q^{71} +4.00000i q^{73} +2.00000i q^{77} +1.00000 q^{81} -12.0000i q^{83} +6.00000i q^{87} -2.00000 q^{89} +4.00000 q^{91} +2.00000i q^{93} +8.00000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 4 q^{11} - 4 q^{19} - 2 q^{21} - 12 q^{29} - 4 q^{31} + 8 q^{39} - 20 q^{41} - 2 q^{49} - 4 q^{51} + 16 q^{59} - 4 q^{61} + 8 q^{69} - 20 q^{71} + 2 q^{81} - 4 q^{89} + 8 q^{91} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^11 - 4 * q^19 - 2 * q^21 - 12 * q^29 - 4 * q^31 + 8 * q^39 - 20 * q^41 - 2 * q^49 - 4 * q^51 + 16 * q^59 - 4 * q^61 + 8 * q^69 - 20 * q^71 + 2 * q^81 - 4 * q^89 + 8 * q^91 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.00000i	− 0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$	− 8.00000i	− 0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.00000i	− 0.601317i
$178$	0	0
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	10.0000i	0.685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	4.00000i	0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$	4.00000i	0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	0	0
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.00000i	0.122398i
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.0000i	0.590281i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.0000i	− 0.774202i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.00000i	0.105851i
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0000i	1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$	20.0000i	1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000i	0.991363i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.00000i	0.0979404i
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	− 8.00000i	− 0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	24.0000i	1.15337i	0.816968	+	0.576683i	$0.195653\pi$
$433$	24.0000i	1.15337i	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.0000i	0.851371i
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	0	0
$453$	4.00000i	0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	−8.00000	−0.368621
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	− 4.00000i	− 0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.0000i	0.448561i
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	− 2.00000i	− 0.0883022i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	− 40.0000i	− 1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$	− 40.0000i	− 1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000i	0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.0000i	0.949370i
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	22.0000i	0.944110i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.0000i	− 1.69485i	−0.530912	−	0.847427i	$-0.678150\pi$
$557$	− 40.0000i	− 1.69485i	0.530912	−	0.847427i	$-0.321850\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	22.0000i	0.919063i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.0000i	− 0.409273i
$598$	0	0
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.0000i	0.805170i	0.915383	+	0.402585i	$0.131888\pi$
$617$	20.0000i	0.805170i	−0.915383	+	0.402585i	$0.868112\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	2.00000i	0.0801283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.00000i	− 0.159745i
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	44.0000i	1.72185i	0.508729	+	0.860927i	$0.330115\pi$
$653$	44.0000i	1.72185i	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	38.0000	1.48027	0.740135	−	0.672458i	$-0.234762\pi$
$659$	38.0000	1.48027	0.740135	+	0.672458i	$0.234762\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 26.0000i	− 0.991962i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.0000i	0.757554i
$698$	0	0
$699$	4.00000	0.151294
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	20.0000i	0.754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.0000i	0.522840i
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 30.0000i	− 1.11571i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	12.0000i	0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$	12.0000i	0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	− 16.0000i	− 0.583072i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.0000i	− 0.508839i	−0.967094	−	0.254419i	$-0.918116\pi$
$757$	− 14.0000i	− 0.508839i	0.967094	−	0.254419i	$-0.0818843\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 14.0000i	− 0.506834i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.0000i	1.15545i
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.0000i	0.358748i
$778$	0	0
$779$	20.0000	0.716574
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.0000i	0.633630i
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−54.0000	−1.89620	−0.948098	−	0.317978i	$-0.896996\pi$
$811$	−54.0000	−1.89620	−0.948098	+	0.317978i	$0.896996\pi$
$812$	0	0
$813$	6.00000i	0.210429i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 24.0000i	− 0.839654i
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.00000i	− 0.0691301i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	26.0000i	0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000i	0.240523i
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	40.0000	1.37118
$852$	0	0
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.0000i	1.16142i	0.814111	+	0.580709i	$0.197225\pi$
$857$	34.0000i	1.16142i	−0.814111	+	0.580709i	$0.802775\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	0	0
$861$	10.0000	0.340799
$862$	0	0
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	−30.0000	−1.01187
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 12.0000i	− 0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	24.0000i	0.794284i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	− 40.0000i	− 1.31662i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.0000i	0.392023i	0.980602	+	0.196011i	$0.0627990\pi$
$937$	12.0000i	0.392023i	−0.980602	+	0.196011i	$0.937201\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	36.0000i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$	36.0000i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.0000i	− 0.387905i
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	2.00000i	0.0641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.k.c.1849.1		2
3.2	odd	2		6300.2.k.m.6049.1		2
5.2	odd	4		420.2.a.b.1.1	✓	1
5.3	odd	4		2100.2.a.k.1.1		1
5.4	even	2	inner	2100.2.k.c.1849.2		2
15.2	even	4		1260.2.a.e.1.1		1
15.8	even	4		6300.2.a.l.1.1		1
15.14	odd	2		6300.2.k.m.6049.2		2
20.3	even	4		8400.2.a.bc.1.1		1
20.7	even	4		1680.2.a.r.1.1		1
35.2	odd	12		2940.2.q.k.361.1		2
35.12	even	12		2940.2.q.g.361.1		2
35.17	even	12		2940.2.q.g.961.1		2
35.27	even	4		2940.2.a.g.1.1		1
35.32	odd	12		2940.2.q.k.961.1		2
40.27	even	4		6720.2.a.d.1.1		1
40.37	odd	4		6720.2.a.bt.1.1		1
60.47	odd	4		5040.2.a.c.1.1		1
105.62	odd	4		8820.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.a.b.1.1	✓	1	5.2	odd	4
1260.2.a.e.1.1		1	15.2	even	4
1680.2.a.r.1.1		1	20.7	even	4
2100.2.a.k.1.1		1	5.3	odd	4
2100.2.k.c.1849.1		2	1.1	even	1	trivial
2100.2.k.c.1849.2		2	5.4	even	2	inner
2940.2.a.g.1.1		1	35.27	even	4
2940.2.q.g.361.1		2	35.12	even	12
2940.2.q.g.961.1		2	35.17	even	12
2940.2.q.k.361.1		2	35.2	odd	12
2940.2.q.k.961.1		2	35.32	odd	12
5040.2.a.c.1.1		1	60.47	odd	4
6300.2.a.l.1.1		1	15.8	even	4
6300.2.k.m.6049.1		2	3.2	odd	2
6300.2.k.m.6049.2		2	15.14	odd	2
6720.2.a.d.1.1		1	40.27	even	4
6720.2.a.bt.1.1		1	40.37	odd	4
8400.2.a.bc.1.1		1	20.3	even	4
8820.2.a.x.1.1		1	105.62	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +4.00000i q^{13} -2.00000i q^{17} -2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} +2.00000i q^{33} -10.0000i q^{37} +4.00000 q^{39} -10.0000 q^{41} +12.0000i q^{43} +8.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +2.00000i q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +12.0000i q^{67} +4.00000 q^{69} -10.0000 q^{71} +4.00000i q^{73} +2.00000i q^{77} +1.00000 q^{81} -12.0000i q^{83} +6.00000i q^{87} -2.00000 q^{89} +4.00000 q^{91} +2.00000i q^{93} +8.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 4 q^{11} - 4 q^{19} - 2 q^{21} - 12 q^{29} - 4 q^{31} + 8 q^{39} - 20 q^{41} - 2 q^{49} - 4 q^{51} + 16 q^{59} - 4 q^{61} + 8 q^{69} - 20 q^{71} + 2 q^{81} - 4 q^{89} + 8 q^{91} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^11 - 4 * q^19 - 2 * q^21 - 12 * q^29 - 4 * q^31 + 8 * q^39 - 20 * q^41 - 2 * q^49 - 4 * q^51 + 16 * q^59 - 4 * q^61 + 8 * q^69 - 20 * q^71 + 2 * q^81 - 4 * q^89 + 8 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more